Remark 3.4 In particular, when \( \dfrac{ \phi _{n}\left(\alpha _{1} \right)}{\sqrt{n}} f^{\prime }_{l}\left(\phi _{n}\left(\alpha _{1} \right) \right)\rightarrow 0 \left( n\rightarrow \infty \right),\)  Theorem REF  is inline with Theorem 2.1 in [1]}, and when \( \dfrac{ \phi _{n}\left(\alpha _{1} \right)}{\sqrt{n}} f^{\prime }_{l}\left(\phi _{n}\left(\alpha _{1} \right) \right)\rightarrow \infty \left( n\rightarrow \infty \right),\)  Theorem REF  is in accord with the CLT derived by [2]}. Moreover, \( \dfrac{\phi _{n}\left(\alpha _{k} \right)}{\sqrt{n}}f_{l}^{\prime }\left( \phi _{n}\left(\alpha _{k} \right)\right) \)  is obtained from the spiked part of the linear spectrum statistics \( \sum _{j=1}^{M}f_{l}\left(\lambda _{j} \right)-\sum _{k=1}^{K}\sum _{j\in J_{k}}f_{l}\left(\phi _{n}\left(\alpha _{j} \right) \right) \)  (which is shown in the Section 5).

